# std::beta, std::betaf, std::betal

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< cpp‎ | experimental‎ | special functions

double beta( double x, double y ); float betaf( float x, float y ); long double betal( long double x, long double y ); | (1) |
Promoted beta( Arithmetic x, Arithmetic y ); | (2) |

1) Computes the beta function of x and y.
2) A set of overloads or a function template for all combinations of arguments of arithmetic type not covered by (1). If any argument has integral type, it is cast to double. If any argument is long double, then the return type `Promoted` is also long double, otherwise the return type is always double.

As all special functions, `beta` is only guaranteed to be available in `<cmath>` if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

### Parameters

x, y | - | values of a floating-point or integral type

### Return value

If no errors occur, value of the beta function of x and y, that is ∫1
0tx-1
(1 - t)(y-1)
dt, or, equivalently,

Γ(x)Γ(y)
Γ(x + y)

is returned.

### Error handling

Errors may be reported as specified in math_errhandling.

- If any argument is NaN, NaN is returned and domain error is not reported.
- The function is only required to be defined where both x and y are greater than zero, and is allowed to report a domain error otherwise.

### Notes

Implementations that do not support TR 29124 but support TR 19768, provide this function in the header `tr1/cmath` and namespace `std::tr1`.

An implementation of this function is also available in boost.math.

beta(x, y) equals beta(y, x).

When x and y are positive integers, beta(x, y) equals \(\frac{(x - 1)!(y - 1)!}{(x + y - 1)!}\)

(x - 1)!(y - 1)!
(x + y - 1)!

. Binomial coefficients can be expressed in terms of the beta function: \(\binom{n}{k} = \frac{1}{(n + 1)B(n - k + 1, k + 1)}\)⎛
⎜
⎝n
k⎞
⎟
⎠=

1
(n + 1)Β(n - k + 1, k + 1)

.

### Example

(works as shown with gcc 6.0)

Run this code

    #define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
    #include <cmath>
    #include <iomanip>
    #include <iostream>
    #include <string>

    double binom(int n, int k)
    {
        return 1 / ((n + 1) * std::beta(n - k + 1, k + 1));
    }

    int main()
    {
        std::cout << "Pascal's triangle:\n";
        for (int n = 1; n < 10; ++n)
        {
            std::cout << std::string(20 - n * 2, ' ');
            for (int k = 1; k < n; ++k)
                std::cout << std::setw(3) << binom(n, k) << ' ';
            std::cout << '\n';
        }
    }

Output:

    Pascal's triangle:

                      2
                    3   3
                  4   6   4
                5  10  10   5
              6  15  20  15   6
            7  21  35  35  21   7
          8  28  56  70  56  28   8
        9  36  84 126 126  84  36   9

### See also

tgammatgammaftgammal (C++11)(C++11)(C++11) | gamma function (function)

### External links

Weisstein, Eric W. "Beta Function." From MathWorld--A Wolfram Web Resource.

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